The joint research in this programme will study important aspects—both theoretical as well as applied—of computing with infinite objects. A central aim is laying the grounds for the generation of efficient and verified software in engineering applications.
Computability with finite objects such as the natural numbers is a well-developed subject with first investigations dating back to the first half of the last century. In contrast, interest in computing with infinite objects is much younger. Notions of infinity appear already in classical computability theory, for example, as higher-order functions or as computations that do not terminate; the emphasis of this project, in contrast, is firmly on infinite data such as may be found in all of mainstream mathematics.
A prime example for infinite data is provided by the real numbers, most commonly conceived as infinite sequences of digits. Since the reals are fundamental in mathematics, any attempt to compute objects of mathematical interest has to be based on an implementation of real numbers. While most applications in science and engineering substitute the reals with floating point numbers of fixed finite precision and thus have to deal with truncation and rounding errors, the approach in this project is different: exact real numbers are taken as first-class citizens and while any computation can only exploit a finite portion of its input in finite time, increased precision is always available by continuing the computation process. We will refer to this mode of computing with real numbers as exact real arithmetic or ERA. These ideas are greatly generalised in Weihrauch’s type-two theory of effectivity (TTE) which aims to represent infinite data of any kind as streams of finite data.
This project aims to bring together the expertise of specialists in mathematics, logic, and computer science to push the frontiers of our theoretical and practical understanding of computing with infinite objects. Three overarching motivations drive the proposed collaboration:
Representation and representability. Elementary cardinality considerations tell us that it is not possible to represent arbitrary mathematical objects in a way that is accessible to computation. We will enlist expertise in topology, logic, and set theory, to address the question of which objects are representable and how they can be represented most efficiently.
Constructivity. Working in a constructive mathematical universe can greatly enhance our understanding of the link between computation and mathematical structure. Not only informs us which are the objects of relevance, it also allows us to devise algorithms from proofs.
Efficient implementation. The project also aims to make progress on concrete implementations. Theoretical insights from elsewhere will thus be tested in actual computer systems, while obstacles encountered in the latter will inform the direction of mathematical investigation.
The specific research objectives of this project are grouped together in three work packages:
WP1 — Foundations
WP2 — Exact Computation in real analysis
WP3 — Logical representation of data
Contact: Dieter Spreen (firstname.lastname@example.org)